We present mid-infrared spectroscopy of a sample of 16 optically faint infrared luminous galaxies obtained with the Infrared Spectrograph (IRS) on the Spitzer Space Telescope. These sources were jointly selected from Spitzer and Chandra imaging surveys in the NDWFS Bootes field and were selected from their bright X-ray fluxes to host luminous AGN. None of the spectra show significant emission from polycyclic aromatic hydrocarbons (PAHs; 6.2um equivalent widths <0.2um), consistent with their infrared emission being dominated by AGN. Nine of the X-ray sources show 9.7um silicate absorption features. Their redshifts are in the range 0.9<z<2.6, implying infrared luminosities of log(L{IR})=12.5-13.6 solar luminosities. The average silicate absorption strength is not as strong as that of previously targeted optically faint infrared luminous galaxies with similar mid-infrared luminosities implying that the X-ray selection favors sources behind a smaller column of Si-rich dust than non-X-ray selection. Seven of the X-ray sources have featureless power-law mid-IR spectra. We argue that the featureless spectra likely result from the sources having weak or absent silicate and PAH features rather than the sources lying at higher redshifts where these features are shifted out of the IRS spectral window. We investigate whether there are any correlations between X-ray and infrared properties and find that sources with silicate absorption features tend to have fainter X-ray fluxes and harder X-ray spectra, indicating a weak relation between the amount of silicate absorption and column density of X-ray-absorbing gas.
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